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## G = Q16.4D4order 128 = 27

### 1st non-split extension by Q16 of D4 acting via D4/C4=C2

p-group, metabelian, nilpotent (class 4), monomial

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C2×C8 — Q16.4D4
 Chief series C1 — C2 — C4 — C8 — C2×C8 — C2×Q16 — C4×Q16 — Q16.4D4
 Lower central C1 — C2 — C4 — C2×C8 — Q16.4D4
 Upper central C1 — C22 — C42 — C4×C8 — Q16.4D4
 Jennings C1 — C2 — C2 — C2 — C2 — C4 — C4 — C2×C8 — Q16.4D4

Generators and relations for Q16.4D4
G = < a,b,c,d | a8=c4=1, b2=d2=a4, bab-1=dad-1=a-1, ac=ca, bc=cb, dbd-1=a-1b, dcd-1=c-1 >

Subgroups: 172 in 77 conjugacy classes, 34 normal (22 characteristic)
C1, C2 [×3], C4 [×2], C4 [×2], C4 [×6], C22, C8 [×2], C8, C2×C4 [×3], C2×C4 [×4], Q8 [×7], C16 [×2], C42, C42, C4⋊C4 [×4], C2×C8 [×2], Q16 [×2], Q16 [×7], C2×Q8 [×3], C4×C8, Q8⋊C4, C2.D8, C2×C16 [×2], Q32 [×4], C4×Q8, C4⋊Q8, C2×Q16, C2×Q16 [×2], C2×Q16, C2.Q32 [×2], C4⋊C16, C4×Q16, C4⋊Q16, C2×Q32 [×2], Q16.4D4
Quotients: C1, C2 [×7], C22 [×7], D4 [×4], C23, D8 [×2], C2×D4 [×2], C4○D4, Q32 [×2], C4⋊D4, C2×D8, C8⋊C22, C4⋊D8, C2×Q32, Q32⋊C2, Q16.4D4

Character table of Q16.4D4

 class 1 2A 2B 2C 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 4K 8A 8B 8C 8D 8E 8F 16A 16B 16C 16D 16E 16F 16G 16H size 1 1 1 1 2 2 2 2 4 8 8 8 8 16 16 2 2 2 2 4 4 4 4 4 4 4 4 4 4 ρ1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 trivial ρ2 1 1 1 1 1 1 1 1 1 1 1 1 1 -1 -1 1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 -1 -1 linear of order 2 ρ3 1 1 1 1 1 -1 -1 1 -1 -1 -1 1 1 -1 1 1 1 1 1 -1 -1 -1 1 1 1 -1 -1 -1 1 linear of order 2 ρ4 1 1 1 1 1 -1 -1 1 -1 -1 -1 1 1 1 -1 1 1 1 1 -1 -1 1 -1 -1 -1 1 1 1 -1 linear of order 2 ρ5 1 1 1 1 1 -1 -1 1 -1 1 1 -1 -1 -1 1 1 1 1 1 -1 -1 1 -1 -1 -1 1 1 1 -1 linear of order 2 ρ6 1 1 1 1 1 -1 -1 1 -1 1 1 -1 -1 1 -1 1 1 1 1 -1 -1 -1 1 1 1 -1 -1 -1 1 linear of order 2 ρ7 1 1 1 1 1 1 1 1 1 -1 -1 -1 -1 1 1 1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 -1 -1 linear of order 2 ρ8 1 1 1 1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 linear of order 2 ρ9 2 -2 -2 2 2 0 0 -2 0 2 -2 0 0 0 0 2 -2 2 -2 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ10 2 -2 -2 2 2 0 0 -2 0 -2 2 0 0 0 0 2 -2 2 -2 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ11 2 2 2 2 2 2 2 2 2 0 0 0 0 0 0 -2 -2 -2 -2 -2 -2 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ12 2 2 2 2 2 -2 -2 2 -2 0 0 0 0 0 0 -2 -2 -2 -2 2 2 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ13 2 2 2 2 -2 -2 -2 -2 2 0 0 0 0 0 0 0 0 0 0 0 0 √2 -√2 √2 -√2 -√2 √2 -√2 √2 orthogonal lifted from D8 ρ14 2 2 2 2 -2 2 2 -2 -2 0 0 0 0 0 0 0 0 0 0 0 0 √2 √2 -√2 √2 -√2 √2 -√2 -√2 orthogonal lifted from D8 ρ15 2 2 2 2 -2 2 2 -2 -2 0 0 0 0 0 0 0 0 0 0 0 0 -√2 -√2 √2 -√2 √2 -√2 √2 √2 orthogonal lifted from D8 ρ16 2 2 2 2 -2 -2 -2 -2 2 0 0 0 0 0 0 0 0 0 0 0 0 -√2 √2 -√2 √2 √2 -√2 √2 -√2 orthogonal lifted from D8 ρ17 2 2 -2 -2 0 -2 2 0 0 0 0 0 0 0 0 -√2 -√2 √2 √2 -√2 √2 ζ165-ζ163 ζ165-ζ163 -ζ167+ζ16 -ζ165+ζ163 ζ167-ζ16 -ζ165+ζ163 -ζ167+ζ16 ζ167-ζ16 symplectic lifted from Q32, Schur index 2 ρ18 2 2 -2 -2 0 -2 2 0 0 0 0 0 0 0 0 -√2 -√2 √2 √2 -√2 √2 -ζ165+ζ163 -ζ165+ζ163 ζ167-ζ16 ζ165-ζ163 -ζ167+ζ16 ζ165-ζ163 ζ167-ζ16 -ζ167+ζ16 symplectic lifted from Q32, Schur index 2 ρ19 2 2 -2 -2 0 2 -2 0 0 0 0 0 0 0 0 √2 √2 -√2 -√2 -√2 √2 -ζ167+ζ16 ζ167-ζ16 ζ165-ζ163 -ζ167+ζ16 ζ165-ζ163 ζ167-ζ16 -ζ165+ζ163 -ζ165+ζ163 symplectic lifted from Q32, Schur index 2 ρ20 2 2 -2 -2 0 -2 2 0 0 0 0 0 0 0 0 √2 √2 -√2 -√2 √2 -√2 -ζ167+ζ16 -ζ167+ζ16 -ζ165+ζ163 ζ167-ζ16 ζ165-ζ163 ζ167-ζ16 -ζ165+ζ163 ζ165-ζ163 symplectic lifted from Q32, Schur index 2 ρ21 2 2 -2 -2 0 2 -2 0 0 0 0 0 0 0 0 -√2 -√2 √2 √2 √2 -√2 -ζ165+ζ163 ζ165-ζ163 -ζ167+ζ16 -ζ165+ζ163 -ζ167+ζ16 ζ165-ζ163 ζ167-ζ16 ζ167-ζ16 symplectic lifted from Q32, Schur index 2 ρ22 2 2 -2 -2 0 2 -2 0 0 0 0 0 0 0 0 -√2 -√2 √2 √2 √2 -√2 ζ165-ζ163 -ζ165+ζ163 ζ167-ζ16 ζ165-ζ163 ζ167-ζ16 -ζ165+ζ163 -ζ167+ζ16 -ζ167+ζ16 symplectic lifted from Q32, Schur index 2 ρ23 2 2 -2 -2 0 -2 2 0 0 0 0 0 0 0 0 √2 √2 -√2 -√2 √2 -√2 ζ167-ζ16 ζ167-ζ16 ζ165-ζ163 -ζ167+ζ16 -ζ165+ζ163 -ζ167+ζ16 ζ165-ζ163 -ζ165+ζ163 symplectic lifted from Q32, Schur index 2 ρ24 2 2 -2 -2 0 2 -2 0 0 0 0 0 0 0 0 √2 √2 -√2 -√2 -√2 √2 ζ167-ζ16 -ζ167+ζ16 -ζ165+ζ163 ζ167-ζ16 -ζ165+ζ163 -ζ167+ζ16 ζ165-ζ163 ζ165-ζ163 symplectic lifted from Q32, Schur index 2 ρ25 2 -2 -2 2 2 0 0 -2 0 0 0 -2i 2i 0 0 -2 2 -2 2 0 0 0 0 0 0 0 0 0 0 complex lifted from C4○D4 ρ26 2 -2 -2 2 2 0 0 -2 0 0 0 2i -2i 0 0 -2 2 -2 2 0 0 0 0 0 0 0 0 0 0 complex lifted from C4○D4 ρ27 4 -4 -4 4 -4 0 0 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from C8⋊C22 ρ28 4 -4 4 -4 0 0 0 0 0 0 0 0 0 0 0 2√2 -2√2 -2√2 2√2 0 0 0 0 0 0 0 0 0 0 symplectic lifted from Q32⋊C2, Schur index 2 ρ29 4 -4 4 -4 0 0 0 0 0 0 0 0 0 0 0 -2√2 2√2 2√2 -2√2 0 0 0 0 0 0 0 0 0 0 symplectic lifted from Q32⋊C2, Schur index 2

Smallest permutation representation of Q16.4D4
Regular action on 128 points
Generators in S128
```(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)(17 18 19 20 21 22 23 24)(25 26 27 28 29 30 31 32)(33 34 35 36 37 38 39 40)(41 42 43 44 45 46 47 48)(49 50 51 52 53 54 55 56)(57 58 59 60 61 62 63 64)(65 66 67 68 69 70 71 72)(73 74 75 76 77 78 79 80)(81 82 83 84 85 86 87 88)(89 90 91 92 93 94 95 96)(97 98 99 100 101 102 103 104)(105 106 107 108 109 110 111 112)(113 114 115 116 117 118 119 120)(121 122 123 124 125 126 127 128)
(1 25 5 29)(2 32 6 28)(3 31 7 27)(4 30 8 26)(9 19 13 23)(10 18 14 22)(11 17 15 21)(12 24 16 20)(33 59 37 63)(34 58 38 62)(35 57 39 61)(36 64 40 60)(41 51 45 55)(42 50 46 54)(43 49 47 53)(44 56 48 52)(65 90 69 94)(66 89 70 93)(67 96 71 92)(68 95 72 91)(73 82 77 86)(74 81 78 85)(75 88 79 84)(76 87 80 83)(97 122 101 126)(98 121 102 125)(99 128 103 124)(100 127 104 123)(105 114 109 118)(106 113 110 117)(107 120 111 116)(108 119 112 115)
(1 43 11 35)(2 44 12 36)(3 45 13 37)(4 46 14 38)(5 47 15 39)(6 48 16 40)(7 41 9 33)(8 42 10 34)(17 57 25 49)(18 58 26 50)(19 59 27 51)(20 60 28 52)(21 61 29 53)(22 62 30 54)(23 63 31 55)(24 64 32 56)(65 97 73 105)(66 98 74 106)(67 99 75 107)(68 100 76 108)(69 101 77 109)(70 102 78 110)(71 103 79 111)(72 104 80 112)(81 113 89 121)(82 114 90 122)(83 115 91 123)(84 116 92 124)(85 117 93 125)(86 118 94 126)(87 119 95 127)(88 120 96 128)
(1 105 5 109)(2 112 6 108)(3 111 7 107)(4 110 8 106)(9 99 13 103)(10 98 14 102)(11 97 15 101)(12 104 16 100)(17 123 21 127)(18 122 22 126)(19 121 23 125)(20 128 24 124)(25 115 29 119)(26 114 30 118)(27 113 31 117)(28 120 32 116)(33 67 37 71)(34 66 38 70)(35 65 39 69)(36 72 40 68)(41 75 45 79)(42 74 46 78)(43 73 47 77)(44 80 48 76)(49 83 53 87)(50 82 54 86)(51 81 55 85)(52 88 56 84)(57 91 61 95)(58 90 62 94)(59 89 63 93)(60 96 64 92)```

`G:=sub<Sym(128)| (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24)(25,26,27,28,29,30,31,32)(33,34,35,36,37,38,39,40)(41,42,43,44,45,46,47,48)(49,50,51,52,53,54,55,56)(57,58,59,60,61,62,63,64)(65,66,67,68,69,70,71,72)(73,74,75,76,77,78,79,80)(81,82,83,84,85,86,87,88)(89,90,91,92,93,94,95,96)(97,98,99,100,101,102,103,104)(105,106,107,108,109,110,111,112)(113,114,115,116,117,118,119,120)(121,122,123,124,125,126,127,128), (1,25,5,29)(2,32,6,28)(3,31,7,27)(4,30,8,26)(9,19,13,23)(10,18,14,22)(11,17,15,21)(12,24,16,20)(33,59,37,63)(34,58,38,62)(35,57,39,61)(36,64,40,60)(41,51,45,55)(42,50,46,54)(43,49,47,53)(44,56,48,52)(65,90,69,94)(66,89,70,93)(67,96,71,92)(68,95,72,91)(73,82,77,86)(74,81,78,85)(75,88,79,84)(76,87,80,83)(97,122,101,126)(98,121,102,125)(99,128,103,124)(100,127,104,123)(105,114,109,118)(106,113,110,117)(107,120,111,116)(108,119,112,115), (1,43,11,35)(2,44,12,36)(3,45,13,37)(4,46,14,38)(5,47,15,39)(6,48,16,40)(7,41,9,33)(8,42,10,34)(17,57,25,49)(18,58,26,50)(19,59,27,51)(20,60,28,52)(21,61,29,53)(22,62,30,54)(23,63,31,55)(24,64,32,56)(65,97,73,105)(66,98,74,106)(67,99,75,107)(68,100,76,108)(69,101,77,109)(70,102,78,110)(71,103,79,111)(72,104,80,112)(81,113,89,121)(82,114,90,122)(83,115,91,123)(84,116,92,124)(85,117,93,125)(86,118,94,126)(87,119,95,127)(88,120,96,128), (1,105,5,109)(2,112,6,108)(3,111,7,107)(4,110,8,106)(9,99,13,103)(10,98,14,102)(11,97,15,101)(12,104,16,100)(17,123,21,127)(18,122,22,126)(19,121,23,125)(20,128,24,124)(25,115,29,119)(26,114,30,118)(27,113,31,117)(28,120,32,116)(33,67,37,71)(34,66,38,70)(35,65,39,69)(36,72,40,68)(41,75,45,79)(42,74,46,78)(43,73,47,77)(44,80,48,76)(49,83,53,87)(50,82,54,86)(51,81,55,85)(52,88,56,84)(57,91,61,95)(58,90,62,94)(59,89,63,93)(60,96,64,92)>;`

`G:=Group( (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24)(25,26,27,28,29,30,31,32)(33,34,35,36,37,38,39,40)(41,42,43,44,45,46,47,48)(49,50,51,52,53,54,55,56)(57,58,59,60,61,62,63,64)(65,66,67,68,69,70,71,72)(73,74,75,76,77,78,79,80)(81,82,83,84,85,86,87,88)(89,90,91,92,93,94,95,96)(97,98,99,100,101,102,103,104)(105,106,107,108,109,110,111,112)(113,114,115,116,117,118,119,120)(121,122,123,124,125,126,127,128), (1,25,5,29)(2,32,6,28)(3,31,7,27)(4,30,8,26)(9,19,13,23)(10,18,14,22)(11,17,15,21)(12,24,16,20)(33,59,37,63)(34,58,38,62)(35,57,39,61)(36,64,40,60)(41,51,45,55)(42,50,46,54)(43,49,47,53)(44,56,48,52)(65,90,69,94)(66,89,70,93)(67,96,71,92)(68,95,72,91)(73,82,77,86)(74,81,78,85)(75,88,79,84)(76,87,80,83)(97,122,101,126)(98,121,102,125)(99,128,103,124)(100,127,104,123)(105,114,109,118)(106,113,110,117)(107,120,111,116)(108,119,112,115), (1,43,11,35)(2,44,12,36)(3,45,13,37)(4,46,14,38)(5,47,15,39)(6,48,16,40)(7,41,9,33)(8,42,10,34)(17,57,25,49)(18,58,26,50)(19,59,27,51)(20,60,28,52)(21,61,29,53)(22,62,30,54)(23,63,31,55)(24,64,32,56)(65,97,73,105)(66,98,74,106)(67,99,75,107)(68,100,76,108)(69,101,77,109)(70,102,78,110)(71,103,79,111)(72,104,80,112)(81,113,89,121)(82,114,90,122)(83,115,91,123)(84,116,92,124)(85,117,93,125)(86,118,94,126)(87,119,95,127)(88,120,96,128), (1,105,5,109)(2,112,6,108)(3,111,7,107)(4,110,8,106)(9,99,13,103)(10,98,14,102)(11,97,15,101)(12,104,16,100)(17,123,21,127)(18,122,22,126)(19,121,23,125)(20,128,24,124)(25,115,29,119)(26,114,30,118)(27,113,31,117)(28,120,32,116)(33,67,37,71)(34,66,38,70)(35,65,39,69)(36,72,40,68)(41,75,45,79)(42,74,46,78)(43,73,47,77)(44,80,48,76)(49,83,53,87)(50,82,54,86)(51,81,55,85)(52,88,56,84)(57,91,61,95)(58,90,62,94)(59,89,63,93)(60,96,64,92) );`

`G=PermutationGroup([(1,2,3,4,5,6,7,8),(9,10,11,12,13,14,15,16),(17,18,19,20,21,22,23,24),(25,26,27,28,29,30,31,32),(33,34,35,36,37,38,39,40),(41,42,43,44,45,46,47,48),(49,50,51,52,53,54,55,56),(57,58,59,60,61,62,63,64),(65,66,67,68,69,70,71,72),(73,74,75,76,77,78,79,80),(81,82,83,84,85,86,87,88),(89,90,91,92,93,94,95,96),(97,98,99,100,101,102,103,104),(105,106,107,108,109,110,111,112),(113,114,115,116,117,118,119,120),(121,122,123,124,125,126,127,128)], [(1,25,5,29),(2,32,6,28),(3,31,7,27),(4,30,8,26),(9,19,13,23),(10,18,14,22),(11,17,15,21),(12,24,16,20),(33,59,37,63),(34,58,38,62),(35,57,39,61),(36,64,40,60),(41,51,45,55),(42,50,46,54),(43,49,47,53),(44,56,48,52),(65,90,69,94),(66,89,70,93),(67,96,71,92),(68,95,72,91),(73,82,77,86),(74,81,78,85),(75,88,79,84),(76,87,80,83),(97,122,101,126),(98,121,102,125),(99,128,103,124),(100,127,104,123),(105,114,109,118),(106,113,110,117),(107,120,111,116),(108,119,112,115)], [(1,43,11,35),(2,44,12,36),(3,45,13,37),(4,46,14,38),(5,47,15,39),(6,48,16,40),(7,41,9,33),(8,42,10,34),(17,57,25,49),(18,58,26,50),(19,59,27,51),(20,60,28,52),(21,61,29,53),(22,62,30,54),(23,63,31,55),(24,64,32,56),(65,97,73,105),(66,98,74,106),(67,99,75,107),(68,100,76,108),(69,101,77,109),(70,102,78,110),(71,103,79,111),(72,104,80,112),(81,113,89,121),(82,114,90,122),(83,115,91,123),(84,116,92,124),(85,117,93,125),(86,118,94,126),(87,119,95,127),(88,120,96,128)], [(1,105,5,109),(2,112,6,108),(3,111,7,107),(4,110,8,106),(9,99,13,103),(10,98,14,102),(11,97,15,101),(12,104,16,100),(17,123,21,127),(18,122,22,126),(19,121,23,125),(20,128,24,124),(25,115,29,119),(26,114,30,118),(27,113,31,117),(28,120,32,116),(33,67,37,71),(34,66,38,70),(35,65,39,69),(36,72,40,68),(41,75,45,79),(42,74,46,78),(43,73,47,77),(44,80,48,76),(49,83,53,87),(50,82,54,86),(51,81,55,85),(52,88,56,84),(57,91,61,95),(58,90,62,94),(59,89,63,93),(60,96,64,92)])`

Matrix representation of Q16.4D4 in GL4(𝔽17) generated by

 14 3 0 0 14 14 0 0 0 0 16 0 0 0 0 16
,
 7 16 0 0 16 10 0 0 0 0 11 15 0 0 9 6
,
 1 0 0 0 0 1 0 0 0 0 10 9 0 0 2 7
,
 0 4 0 0 4 0 0 0 0 0 10 9 0 0 6 7
`G:=sub<GL(4,GF(17))| [14,14,0,0,3,14,0,0,0,0,16,0,0,0,0,16],[7,16,0,0,16,10,0,0,0,0,11,9,0,0,15,6],[1,0,0,0,0,1,0,0,0,0,10,2,0,0,9,7],[0,4,0,0,4,0,0,0,0,0,10,6,0,0,9,7] >;`

Q16.4D4 in GAP, Magma, Sage, TeX

`Q_{16}._4D_4`
`% in TeX`

`G:=Group("Q16.4D4");`
`// GroupNames label`

`G:=SmallGroup(128,941);`
`// by ID`

`G=gap.SmallGroup(128,941);`
`# by ID`

`G:=PCGroup([7,-2,2,2,-2,2,-2,-2,448,141,64,422,352,1684,438,242,4037,1027,124]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^8=c^4=1,b^2=d^2=a^4,b*a*b^-1=d*a*d^-1=a^-1,a*c=c*a,b*c=c*b,d*b*d^-1=a^-1*b,d*c*d^-1=c^-1>;`
`// generators/relations`

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